平面常应力弹性固支板基波频率映射解析

基于含平面常应力的复杂域弹性固支板基波频率的求解问题,应用共形映射数值理论,将复杂边界域分为奇数与偶数插值点,提出三角插值法和法向收敛法,求解了复杂域与单位圆相互转换的共形映射函数,将其与Galerkin的方法相结合,可完成含平面常应力的复杂板域的振动微分方程及基波频率解析。同时以含圆角的矩形弹性固支板振动微分方程与基波频率求解为示例,分别分析了边长比λ、面积A和常应力系数Sp对基波频率的影响。     

振动固支弹性板基波频率共形解析

基于振动固支弹性异型板基波频率值的求解问题,采用共形映射理论,为了寻求复杂域与单位圆域的共形映射函数,将复杂边界域插值点序列分为奇偶两序列并相互迭代;提出了三角插值数学方法及其法向收敛方法,求得了共形映射函数的复系数。应用Galerkin的方法,完成复杂振动板域振动微分方程的基波频率解析。以椭圆板域求解为示例,分析了偏心率e和面积对基波频率系数的影响。     

多集中质量离散固支弹性板基波频率解析

针对含集中质量的离散固支弹性板基波频率的求解问题,采用小挠度弹性板变形理论与Rayleigh-Ritz能量法,建立含多集中质量弹性板的振动能量方程及其模态方程。结合弹性板的离散点固支边界条件,完成离散固支弹性板的基波振型函数的解析描述,并利用能量极值法,完成随着集中质量大小与位置变化的离散固支弹性板基波振型函数插值系数及其基波频率的解析。以不含集中质量离散固支弹性板为特例求解,与有限元分析计算结果相比对,验证求解离散固支弹性板基波频率方法的可行性。此外,以含两个集中质量的矩形弹性板为计算实例,完成了集中质量大小以及位置变化的基波振型函数插值系数k及其振型函数、基波频率f的曲面描述与分析。上述解析分析方法可为工程电路板PCB的动态设计提供了可行的理论分析方法。     

四边弹性约束FGM矩形板面内自由振动的DQM求解

假设矩形板为正交各向异性,材料的物性沿矩形板的宽度方向按幂律连续分布,基于二维线弹性理论,建立了四边弹性约束功能梯度材料(Functionally Graded Material,FGM)矩形板面内自由振动的控制偏微分方程。控制方程为复杂耦合的变系数偏微分方程,采用微分求积法(Differential Quadrature Method,DQM)数值研究了四边弹性约束FGM矩形板面内自由振动的无量纲频率特性。通过设置弹性刚度系数为0或∞,梯度指数为0,问题退化为各种典型边界下矩形板的面内自由振动,与已有的各向同性矩形板自振频率结果进行比较,结果表明分析求解方法行之有效。最后考虑了FGM矩形板边界条件、长宽比、梯度指数及刚度系数对自振频率的影响。     

四边弹性约束矩形板面内自由振动的DQM求解

基于二维线弹性理论,应用Halmiton原理,建立了四边弹性约束边界矩形板面内自由振动的控制偏微分方程。采用微分求积法(DQM)数值研究了弹性约束边界矩形板面内自由振动的无量纲频率特性。通过设置弹性刚度系数为0或∞,问题退化为各种典型边界矩形板的面内自由振动,与已有的矩形板面内自振频率结果进行比较,结果显示,该分析求解方法行之有效;最后考虑了矩形板边界条件、长宽比、刚度系数对自振频率的影响。     

复杂边界单连通域共形映射解析建模研究

在工程实际中任意非规则复杂边界单连通域问题的数学模型建立难度大，导致求解和优化参数非常困难，根据共形映射原理，采用复变三角插值理论，利用法线迭代收敛方法，将任意复杂边界单连通域问题转化为单位圆域问题，系统地建立了二者间的共形映射函数，并对具有对称轴特性的单连通域问题进行了算法简化。     

异型挤压模腔纵向曲线与共形映射

应用共形映射理论,在异型材非圆截面轮廓上,通过有限奇偶插值点的相互数值求解,建立异型材截面域与单位圆域相互转化的共形映射函数。将异型材挤压成形转化为二维问题,求得塑性流函数,建立各种纵向曲线的模腔曲面数学模型。应用上限原理,对比三种模腔纵向曲线及优化模腔参数,并以椭圆异型材为示例进行优化求解与分析。     

温度影响下Winkler-Pasternak弹性地基上多孔FGM矩形板的自由振动分析

基于经典薄板理论和Hamilton原理研究温度影响下Winkler-Pasternak弹性地基上多孔功能梯度材料(FGM)矩形板的自由振动特性。采用Voigt混合幂率模型和孔隙任意分布模型来表征多孔FGM矩形板的材料属性,并考虑多孔FGM矩形板内部均匀温升和材料具有温度依赖特性;应用物理中面推导弹性地基上多孔FGM矩形板自由振动的控制微分方程并进行无量纲化;采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,引入典型的六种边界在MATLAB统一编程且保证计算精度一致,经过迭代收敛,求解出无量纲固有频率;通过算例研究了边界条件、梯度指数、升温、孔隙率、长宽比、边厚比、无量纲弹性刚度系数和无量纲剪切刚度系数对多孔FGM矩形板振动特性的影响。     

矩形薄板自由振动的一般解析解

本文求得了矩形薄板弹性横向自由振动位型函数微分方程的一般解。可以求解任意边界矩形薄板的固有频率。以四边固定矩形板为例求解了板的基频及其位型。     

携带集中质量的矩形薄板面外非线性动力失稳研究

对于面内对边周期荷载作用下携带集中质量的矩形薄板,当周期荷载的激振频率在板的两倍自振频率附近时,板发生面外参数共振失稳。该文基于薄板大挠度理论,运用伽辽金法推导出携带集中质量的矩形薄板非线性动力失稳的Mathieu-Hill方程,进而求解得到板发生面外参数共振失稳时周期荷载的临界激振频率域以及非线性动力失稳曲线。运用有限元软件进行瞬态分析得到不同激振幅值作用下板发生面外参数共振失稳时周期荷载的最小与最大临界激振频率值,通过与解析解进行对比,验证了计算结果的正确性。研究结果表明:随着集中质量的增加,参数共振失稳的临界激振频率及其不稳定域的宽度逐渐减小,不稳定域的位置逐渐向低激振频率的方向移动;随着集中质量的增加,面外参数共振失稳域的临界激励幅值逐渐增加;随着集中质量所处位置的模态位移增加,不稳定域的宽度减小。     

Material tailoring for orthotropic elastic rotating disks

The tailoring of elastic moduli in the radial direction is studied to design a fiber-reinforced orthotropic linear elastic rotating disk with constant radial or hoop stress or constant in-plane shear stress. For fibers arranged in concentric circles the axes of material symmetry coincide with the radial and the circumferential directions. However, when fibers are aligned along helices, the orientation of material principal axes varies with the radial coordinate of a point. For a solid disk made of an orthotropic material with Young’s moduli proportional to each other, we give explicit expressions for the required variations of the elastic moduli with the radius to attain a given state of stress. For a rotating annular disk composed of a fiber-reinforced composite with fibers placed along concentric circles, the required radial variation of the volume fraction of fibers is calculated numerically and exhibited graphically. For fibers of known volume fraction laid along helices, the radial variation of the fiber orientation angle is determined. We have also analyzed the material tailoring problem for a disk of variable thickness. Results presented herein should help structural engineers and material scientists optimally design rotating disks composed of radially inhomogeneous materials.     

Comments on the pore radius distribution in near-planar stochastic fibre networks

The pore radius distribution in near-planar stochastic fibre networks is known to be influenced by changes in the mean number of fibres per unit area and their distribution in the plane. Experimental data is presented that confirms the established result that the standard deviation of pore radii is proportional to the mean. The data shows also that this proportionality is the same for changes in the number of fibres per unit area and for changes in the uniformity of their in-plane distribution. Data from the literature suggests that processes that increase the mean pore radius, increase also the coefficient of variation of pore radii. Theoretical considerations and experimental data are presented that show that the coefficient of variation of pore radii is in fact constant for near-random and non-random stochastic fibre networks.     

基于细观力学的大开孔层合板缝合补强力学性能计算的新方法基于细观力学的大开孔层合板缝合补强力学性能计算的新方法

从细观力学的角度出发,考虑了面内纤维弯曲及富树脂缺陷,建立了大开孔层合板缝合补强孔边针脚损伤的单胞模型。建立了纤维弯曲函数,推导了纤维弯曲区域的纤维体积分数及纤维弯曲角度。基于复合材料力学分析方法,计算得出了单胞的材料弹性常数。研究表明:缝合导致单胞面内纤维最大弯曲角不超过20°,单层板纵向杨氏模量减小,横向杨氏模量、剪切模量及泊松比均增大,变化幅度均在-8%~20%之间;且对于大开孔层合板缝合补强而言,针距变化引起的材料性能变化相对边距大许多。由上述计算结果,建立了一种缝合补强大开孔层合板力学性能计算的新方法,同时引入针孔模拟针脚处的应力集中现象,结果表明:缝合会造成层合板面内力学性能降低,并且对面内的压缩性能影响大于对面内拉伸性能的影响。      

复合材料夹层板面芯二维分层屈曲研究

本文建立了一个分析复合材料夹层板面芯椭圆分层屈曲的二维弹性基础模型;并找到了一个能很好地反映分层屈曲变形特征的位移表达式.通过应用瑞利-里兹能量法得到了分层的临界屈曲载荷和有效边界刚性系数C'.本文发现,经典的薄膜分层模型不能反映夹层板芯子对面芯分层的横向弹性支持作用,只能给出面芯分层屈曲的上临界载荷.本文还得到了一些分层参数与有效边界刚性系数C'的关系.     

Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a pre-assigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r². The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r². For a hollow cylinder fixed on the inner surface and subjected to tangential tractions on the outer surface (or vice versa) the through-the-thickness in-plane shear stress distribution is also universal and is determined by surface tractions and the outer radius of the cylinder; it is independent of the spatial variation of the shear modulus.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

正交平面织物球面成型渔网模型的几何分析

为了从理论上表征平面正交织物球面成型后的几何特征, 提出基于坐标变换求解渔网模型的新方法, 确定单层正交平面织物在球面上铺覆成型后的网格位置、 织物剪切变形和纱线弯曲变形。依据弧长不变条件确定方形织物完全包覆球面后的对称面上的网格位置和局部坐标系下中间网格的位置, 利用坐标变换获得中间网格在整体坐标系成型球面上的坐标位置; 根据变形前后的网格形状确定织物面内剪切变形和两个方向纱线的弯曲曲率, 为织物的球面成型性评价提供几何参数。通过实例证明了当网格尺寸远小于球体半径时铺覆变形程度与网格 尺寸无关, 也与球体半径无关。铺覆后织物的剪切变形和纱线弯曲变形分布只与织物在球面上的球坐标位置有关。